The slope of a curve passing through the point $(2, 0)$ at $(x, y)$ is $x^2 - 2x$. Find the value of $y$ at $x = 0$.

Let $f:[0,2] \rightarrow R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $F(x)=\int_0^{x^2} f(\sqrt{t}) dt$ for $x \in [0,2]$. If $F'(x)=f'(x)$ for all $x \in (0,2)$,then $F(2)$ equals

The rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant. It is found that the number doubles in $6 \text{ hours}$. The number of bacteria becomes $n$ times the initial number at the end of $18 \text{ hours}$. Find $n$.

Any curve passes through the point $(3, -4)$. If the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2y}{x}$,then the equation of the curve is . . . . . . .

$A$ differential equation for the temperature $T$ of a hot body as a function of time,when it is placed in a bath which is held at a constant temperature of $32^{\circ} F$,is given by (where $k$ is a constant of proportionality):

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} F$ and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$,where $K$ is a positive constant. If $T(15)=120^{\circ} F$,then $T(45)$ is equal to . . . . . . . . (in $^{\circ} F$)